Algebraic cycles and fibrations

For a scheme X over a field k, CHi(X) denotes the rational Chow group of i-dimensional cy-cles on X modulo rational equivalence. Throughout, f: X → B will be a projective surjective morphism defined over k from a quasi-projective variety X of dimension dX to an irreducible quasi-projective variety B of dimension dB, with various extra assumptions which will be explic-itly stated. Let h be the class of a hyperplane section in the Picard group of X. Intersecting with h induces an action CHi(X) → CHi−1(X) still denoted h. Our first observation is Proposition 1.6: when B is smooth, the map dX−dB⊕ i=0 hdX−dB−i ◦ f ∗: dX−dB⊕ i=0 CHl−i(B) − → CHl(X) (1) is injective for all l and a left-inverse can be expressed as a combination of the proper push-forward f∗, the refined pullback f ∗ and intersection with h. It is then not too surprising that